Computational Complexity and other fun stuff in math and computer science from Lance Fortnow and Bill Gasarch

Friday, November 17, 2006

The Future of Science

New Scientist celebrates their 50 years by asking about 70
"Brilliant Minds" to
forecast the next 50 years of science. Several researchers (all well-known
though a few overrated) talk about
math and
computing including
Tim Gowers focusing
on the P versus NP problem.

This problem gets to the heart of mathematics, because mathematical
research itself has the property I have described: it seems to be
easier to check that a proof is correct than to discover it in the
first place. Therefore, if we found a solution to the P = NP problem
it would profoundly affect our understanding of mathematics, and would
rank alongside the famous undecidability results of Kurt Gödel and
Alan Turing.

Well, it follows straightforwardly from Goedel's reasoning that "P=/= NP" is a consequence of the thesis:

An arithmetical relation F(x1, ..., xn) is Turing-computable as 'true' for every n-ary sequence of natural numbers if, and only if, [F(x1, ..., xn)] is a PA-provable formula.

The thesis seems to be intuitively unobjectionable, even though Turing's reasoning indicates that it is neither 'provable' nor 'disprovable' effectively, since we cannot establish, for an arbitrary "F(x1, ..., xn)", that "F(x1, ..., xn) is Turing-computable as 'true' for every n-ary sequence of natural numbers".

So, is it fair to conclude that the resolution of the P versus NP problem would, necessarily, be as profound as, and "rank alongside the famous undecidability results of Kurt Gödel and Alan Turing"?

Most of these forecasts are disappointingly predictable: each scientist interviewed basically explains that in the next 50 years,the whole world will be doing what they have been doing in the last 50 years.One exception is Gowers on P=NP(but if you read Luca Trevisan's paper, or god forbid Gowers's papers, you my have noticed that his work has actually some connections to TCS).